Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 34D / iv

The Maxwell-Boltzmann velocity distribution is unaffected by the position-dependent potential, since the canonical density factors. Its speed density is proportional to $v^2{e}^{-m{v}^2/(2k_{B}T)}$ for $v\ge 0$. For $v>0$, differentiating its logarithm gives $2/v-mv/(k_{B}T)=0$, soThe density vanishes at zero and at infinity and has only this positive stationary point, so it is the maximum. It is the same at every height.